Finite Group Databasebeta

← #31790289

Subgroups — #31790289

Order 17536.

526 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 141 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17984114C211yesmaximal, characteristic, complemented
2#22577970C211yesmaximal, characteristic, complemented
3#22577878C211yesmaximal, characteristic, complemented
4#22577616C214yesmaximal, complemented
5#22577596C214yesmaximal, complemented
6#22577429C211yesmaximal, characteristic, complemented
7#22577408C211yesmaximal, characteristic, complemented
8#22577400C211yesmaximal, characteristic, complemented
9C22×D8×C137C211yesmaximal, characteristic, complemented
10C2×#14555510C2211yescharacteristic
11#17984114—24no—
12C2×#14555498C2212yescomplemented
13C2×#14555495C2212yescomplemented
14C2×#14555494C2211yescharacteristic
15#17984084C2212yescomplemented
16#17984075C2212yescomplemented
17#17984059—22no—
18#17984057—22no—
19D4×Dic137—22no—
20D4×Dic137C2214yescomplemented
21#17984047C2211yescharacteristic
22#17984047C2211yescharacteristic
23C2×C22⋊Dic137—22no—
24#17984017C2214yescomplemented
25#17984013C2212yescomplemented
26#17984010C2212yescomplemented
27#17984009C2214yescomplemented
28C22×D4×C137C2211yescharacteristic
29C22×D4×C137C2211yescharacteristic
30C2×D8×C137—24no—
31C2×D8×C137C2214yes—
32C22×C8×C137C2211yescharacteristic
33#14555510—22no—
34#14555510—22no—
35C2×Dic274C2311yescharacteristic
36C2×D548C2311yescharacteristic
37C2×C22⋊D137—24no—
38C2×C22⋊D137—21no—
39C22×Dic137—24no—
40C2×C4×D137—21no—
41#14555498—24no—
42#14555495—24no—
43#14555494—22no—
44C2×#11764040C2312yes—
45C4⋊Dic137D412yes—
46C4⋊Dic137C2312yes—
47C22⋊Dic137—24no—
48C22⋊Dic137—24no—
49C22⋊Dic137—21no—
50C22⋊Dic137D412yes—
51C4×Dic137—21no—
52C4×Dic137C2312yes—
53C24×C137—21no—
54C24×C137—21no—
55C2×D4×C137—21no—
56C2×D4×C137—22no—
57C2×D4×C137—21no—
58C2×D4×C137—22no—
59C2×D4×C137—22no—
60C2×D4×C137D414yescomplemented
61C2×D4×C137C2312yes—
62C2×D4×C137C2312yes—
63C22×C4×C137C2311yescharacteristic
64D8×C137—28no—
65C2×C8×C137—22no—
66C2×C8×C137C2312yes—
67C22×D137—21no—
68Dic274—21no—
69Dic274—21no—
70D548—21no—
71D548—21no—
72C22⋊D137—42no—
73C22⋊D137—84no—
74C2×Dic137—22no—
75C2×Dic137—21no—
76C2×Dic137—84no—
77C2×Dic137C2×D414yes—
78C4×D137—42no—
79#11764040—22no—
80C23×C137—22no—
81C23×C137—24no—
82C23×C137—22no—
83C23×C137—42no—
84C23×C137—82no—
85C23×C137—82no—
86C23×C137C2×D411yescharacteristic
87D4×C137—24no—
88D4×C137—22no—
89D4×C137—22no—
90D4×C137—22no—
91D4×C137—42no—
92D4×C137C2×D414yes—
93C2×C4×C137C2×D411yescharacteristic
94C2×C4×C137C2×D412yes—
95C2×C4×C137#4112yes—
96C2×C4×C137C2411yescharacteristic
97C8×C137—22no—
98C22×C137—22no—
99C22×C137—24no—
100C22×C137—44no—
101C22×C137—42no—
102C22×C137—44no—
103C22×C137—84no—
104C22×C137—84no—
105C22×C137#12112yes—
106C22×C137C22×D411yescharacteristic
107C2×D137—41no—
108C2×D137—42no—
109C4×C137#12112yes—
110C4×C137C22×D411yescharacteristic
111C4×C137C2×#4111yescharacteristic
112Dic137—24no—
113Dic137—41no—
114Dic137—42no—
115C2×C137—22no—
116C2×C137—44no—
117C2×C137—82no—
118C2×C137C2×#12111yescharacteristic
119C2×C137#57612yes—
120D137—81no—
121C137#304111yescharacteristic, complemented, Sylow
122#3041—1371nomaximal, Sylow
123C2×AGL(1,ℤ/8)—1371no—
124C22×D8D13711yescharacteristic, complemented
125#535—1371no—
126#518—1371no—
127#463—1374no—
128#459—1374no—
129#424—1371no—
130#418—1371no—
131#416—1371no—
132C2×#41—1371no—
133C22×D4C2×D13711yescharacteristic
134C22×D4C2×D13711yescharacteristic
135AGL(1,ℤ/8)—2744no—
136C2×SD16—1372no—
137C2×D8—24no—
138C2×D8—1372no—
139C2×D8C2×D13714yes—
140C2×M16—1371no—
141C22×C8C2×D13711yescharacteristic
142#127—1372no—
143#124—1372no—
144#121—2742no—
145#120—2742no—
146C4×D4—2742no—
147C4×D4—1374no—
148#117—1371no—
149#117—1371no—
150C2×C22⋊C4—2742no—
151C4⋊C8—1374no—
152#103—1372no—
153#102—1374no—
154#102—1372no—
155C24—21no—
156C24—21no—
157#41—2742no—
158#41—2742no—
159C2×Q8—1371no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—22no—
163C2×D4—22no—
164C2×D4—22no—
165C2×D4—2741no—
166C2×D4—2744no—
167C2×D4—1371no—
168C2×D4C22⋊D13714yescomplemented
169C2×D4C22×D13712yes—
170C2×D4C22×D13712yes—
171C22×C4—2741no—
172C22×C4—2744no—
173C22×C4C22×D13711yescharacteristic
174SD16—2744no—
175D8—28no—
176D8—2744no—
177M16—2742no—
178C2×C8—22no—
179C2×C8—1372no—
180C2×C8C22×D13712yes—
181C4⋊C4—1372no—
182C4⋊C4—1372no—
183C22⋊C4—2741no—
184C22⋊C4—2744no—
185C22⋊C4—2744no—
186C22⋊C4—1372no—
187C42—2741no—
188C42—1372no—
189C23—22no—
190C23—24no—
191C23—22no—
192C23—42no—
193C23—82no—
194C23—82no—
195C23—2741no—
196C23D4×D13711yescharacteristic
197Q8—2741no—
198Q8—2741no—
199D4—24no—
200D4—22no—
201D4—22no—
202D4—22no—
203D4—42no—
204D4—2741no—
205D4—2741no—
206D4—5482no—
207D4—10964no—
208D4C2×C22⋊D13714yes—
209C2×C4—2742no—
210C2×C4—2741no—
211C2×C4—5482no—
212C2×C4—10964no—
213C2×C4—1374no—
214C2×C4D4×D13711yescharacteristic
215C2×C4C2×C22⋊D13712yes—
216C2×C4#1455550912yes—
217C2×C4C23×D13711yescharacteristic
218C8—22no—
219C8—2742no—
220C22—22no—
221C22—24no—
222C22—44no—
223C22—44no—
224C22—42no—
225C22—84no—
226C22—84no—
227C22—5482no—
228C22—5481no—
229C22#1798406012yes—
230C22C2×D4×D13711yescharacteristic
231C4—2744no—
232C4—5482no—
233C4—5481no—
234C4#1798406012yes—
235C4C22×C22⋊D13711yescharacteristic
236C4C2×#1455550911yescharacteristic
237C2—22no—
238C2—44no—
239C2—82no—
240C2—10961no—
241C2C2×#1798406011yescharacteristic
242C2#2257820412yes—
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
FittingC22×D8×C137C2
socleC22×C137C22×D4
hypercenterC22×D8D137
hypocenterC137#3041
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#17984114C2semidirect
2. #22577970C2semidirect
3. #22577878C2semidirect
4. #22577616C2semidirect
5. #22577596C2semidirect
6. #22577429C2semidirect
7. #22577408C2semidirect
8. #22577400C2semidirect
9. C22×D8×C137C2semidirect
12. C2×#14555498C22semidirect
13. C2×#14555495C22semidirect
15. #17984084C22semidirect
16. #17984075C22semidirect
20. D4×Dic137C22semidirect
24. #17984017C22semidirect
25. #17984013C22semidirect
26. #17984010C22semidirect
27. #17984009C22semidirect
60. C2×D4×C137D4semidirect
121. C137#3041semidirect
124. C22×D8D137semidirect
168. C2×D4C22⋊D137semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#179841142
#225779702
#225778782
#225776162
#225775962
#225774292
#225774082
#225774002
C22×D8×C1372
#3041137